"For instance, if an exercise was performed in which the basses held a solid low C and the sopranos moved chromatically from a C up the octave (slowly, carefully tuning each note), what would be the intervallic relationship between each note and the pedal C?" - pml quoting Chris

Chris, I would answer this differently from what pml has said. The natural tendency in choirs is to choose intervals that are the least dissonant. This underlies the tendency to sing just intervals in the first place, that is, to eliminate tuning dissonance. C to C#, for example, is not an interval you would normally find in any harmonic structure, but a minor second or major seventh is quite normal in seventh chords. So choirs will naturally choose Db rather than C#. D will be the larger whole step above C, the next note will be the just m3 (Eb), then M3 (E), P4 (F), then +4 (F#, since it is much more closely related to the key of C than Gb, although to be fair, this choice is the most ambiguous of the lot), then P5 (G), then m6 (Ab, since G# is a diminished fourth or +5 with respect to C and much more dissonant and remotely related harmonically), then M6 (A), m7 (Bb), then M7 (B), and finally the octave C.

We can also look at frequency ratios as a criterion, which is just another way of looking at the same tuning phenomenon. In this case, we find that the chromatic half step in the augmented prime in C# over C has a 25/24 frequency ratio, while the diatonic half step in Db over C has a 16/15 frequency ratio. It is clear that the latter represents a significantly simpler harmonic relationship. Complex frequency relationships characterize dissonance if we look at relative dissonance as raw acoustic dissonance as opposed to the more subjective and musically sophisticated use of dissonance in music theoretical contexts in which we are considering how a given harmonic structure tends to resolve rather than any static characteristic of a particular interval. This distinction in the two uses of the concept of dissonance is important, since they are quite different. For example, in the music theoretical concept, the P4 is dissonant merely because it wants to resolve, perhaps with the top note down to an M3, for example, or both out to a major or minor sixth. However, in terms of acoustic dissonance, a P4 is not significantly more dissonant than a P5, which in music theoretical terms is a much more stable interval musically than the P4.

What a great discussion! But a word of caution from the singer's point of view... the "why" of all of this is very interesting from a theoretical point of view, but it can get in the way of good singing if you aren't careful.

You want your heart surgeon to know all about the why and how your heart pumps, and your brain surgeon to know why and how your brain works, but hearts and brains have been working just fine back to our hunter gatherer days when we thought that the liver was the seat of the soul. There is an interesting disconnect... we assume that the audience will hear and respond to chords that aren't in tune, just as we assume that they respond to particularly perfectly well-performed music, but then at the same time we assume that the singers in the choir, who presumably have more innate musicality and interest in getting it right, can't hear and correct tuning issues without a complicated explanation and tuning exercises? Get out of town! I don't doubt the theory at all, but it isn't necessary for you, or your singers, to understand the theory to sing correctly. There are shades of tuning to different chords. As human beings, we hear them, we respond to them viscerally, and if we are singing, we automatically adjust for them, as long as no one gets in our way and asks us to over-think the issue!

In my experience tuning problems happen in the following conditions: 1. Someone isn't thoroughly warmed up yet, and it is physically difficult to control their voice with precision. Think of the voice cracks you get when you're coming down with a cold. Yep. Like that, but to a lesser degree. 2. It's cold or dry in the place where you are singing or someone is wearing perfume or uses obnoxious fabric softener. Sucking a bunch of cold, dry, or allergy-laden air over your vocal chords every time you take proper deep singing breath will tense everything up, dry everything out, or make you cough or crack in extreme situations. Even if it's only a minor annoyance, it makes it hard to control your voice, particularly in parts of your range that are already difficult. People get tempted to go for the full beauty routine for a concert with the smelly soaps and aftershaves. Remind them not to! And make sure your room is warm, or your singers will never be able to warm up. 3. The acoustics in the room aren't helping. The shape of one of the practice rooms where I went to college meant a certain chord would actually make the light fixtures buzz. The acoustics in there also meant that certain chords just sounded dead, so like good singers, we had a natural tendency to compensate. What sounded "correct" to our ears was actually out of tune. What was technically correct sounded incorrect. In that sort of situation, it doesn't really matter that you are technically out of tune, because an audience will be experiencing the same auditory effect that your singers are. (Assuming that the perception of the sound isn't different down at the other end of the room, which is completely possible). If you are singing with a piano, everyone will be pulled between trying to naturally stay in tune with the piano, and trying to naturally adjust for the acoustics, and THEN everyone will be uncomfortable. But a cappella, it will make no difference. Just ignore the fact that you are shifting in and out of tune throughout the piece. If you are ending up so flat the basses are running out of low notes, consider taking the whole piece up a step. 4. Your singers can't hear all the parts in a balanced fashion because you have unequal numbers of singers on each part, or because someone's tone quality is fighting with their neighbor, or someone is just singing really loudly in their neighbor's ear. If no one can hear the basses, for example, then the entire chord shifts. What was a ___ chord, now sounds like a ___ , since it's missing the bass note, and the rest of the choir shifts their tone color and tuning slightly as a result. Like good singers! Who are instinctively trying to stay in tune! The basses, meanwhile, can't hear the sopranos, so to them the ___ chord sounds like a ___, they shift their tuning and tone quality ever so slightly, but in a different direction. Moving people around the room to take advantage of the room's acoustics can help everyone hear each other better. Try switching where the altos stand with where the sopranos stand. Try switching two singers so that they are no longer standing next to each other. Ask parts to mix-up. No one is allowed to stand next to someone on their part, so everyone can hear the full chord progressions. Walk around the room yourself while your singers are singing and listen to what it sounds like from where each of them stands. Adjust where you place them accordingly. When they can hear, the tuning problems will disappear magically. Above all, do not get frustrated and berate your singers for not listening or subject them to lots of tuning exercises. It's a waste of time, and they will get frustrated and indignant with you... because they are listening... and that's why you're having the problem in the first place. Point out the tuning issue. Ask who can't hear each other. Keep your cool and do the best you can. Be prepared to just let it go. 5. You are singing a straining piece of music. I'm an alto, so the typical hymn part for me sits on the D above middle C for the the entire 6 verses and 6 refrains. DDDDDDDDDDDEDDDDDDDDCDDDDD... That's right on the break between my head and chest voice. Great! Says the composer! It's right smack in the middle of the alto range! You must love me so much! No, says the alto! That's really straining, and by the second verse it's hard to keep it in tune because it is physically painful and I am going hoarse! It is not quite comfortable to place it in my chest voice. It is not quite comfortable to place it in my head voice either. Ditto asking someone to sing in the lowest part of their range for a long time, ditto asking someone to sing in the very highest part of their range for a long time. Good singable music gives ALL the vocal parts a chance to move throughout their natural vocal range at least a little bit. 6. Your singers are tired. Going out of tune at the end of rehearsal? Don't let it bother you. Time to stop singing. Send them home to bed. 7. You're all thinking about it too much. If you constantly harp on tuning an obliging singer will overcompensate, or get flustered. 8. They don't know the notes, or the rhythm is tricky, or that page turn is throwing them, or the pronunciation of that latin is getting them. The brain can only handle so many things at once. If the singers are unsure about one aspect of the music, their brain is occupied trying to puzzle it out, and has no time left to pay attention to tuning. Tuning always going in the same spots? Ask if anyone needs to run that spot more slowly with just their part, or speak the words to the rhythm, or run their line against the tenor line. I bet some hands will go up. People get nervous about asking for individual help for their part because they don't want to hold up rehearsal, but it's a better use of time to fix the actual problem than to keep running the same spot without getting any better.

This highly useful discussion has reasonably moved into the huge domain of tuning temperaments. This, together with the astutely informed company here, emboldens me to ask why a "tyranny of the octave" has been uncritically assumed. Laying out the question will not require much math. It'll be about, let's say, intuitive energetics.

In some contrast to assembled company, my own experience directing has all been with amateur a cappella choirs for whom the discussion here might as well be about eye of newt and wing of bat. I have been practicing the "art of the occasionally barely possible", and the occasions have been thrilling and far between.

As to singing intervals, I count arco string playing as singing, and more as well. Fretted instruments allow for niceties of intonation beyond locating tied frets. A note can be bent by other mechanisms--force of stop, nearness to the bridgeward fret, and string-axial finger motion after initial stop. Skilled embouchure enables brass and woodwinds to achieve precision beyond what keying dictates. Woodwinds have also the technique of lifting fingers slightly.

I assume that the human tendency to prefer harmonic intervals is a seamless instance of a widely general tendency, in us and other animals, to find lowest-energy solutions to hundreds of bodily tasks.

To give a musical example, I once had a church choir member, amateur, who developed an idiosyncratic bend of extreme vowels to a cheated value, somewhere between shwa and "ah". This was no expertly cheated high note by an accomplished soprano. I guessed this contralto sensed how she could sustain her invented diphthong with less energy--more comfortably, with more inner satisfaction. The result, repeated at the conclusion of every litany, every Sunday, was "To-ah thee-ah, O-ah Lo-ahd", with smooth but glaring glides.

Now to get closer to my question and thence to an alternative.

Helmholtz (On the Sensations of Tone, Dover) gives a graphic explanation of why P4 and more especially P5 are more confidently performed in tune than any other, within AND INCLUDING the octave, based on overtone combination energetics. He doesn't talk in terms of performance, but couches the demo in terms of how wide a departure from ideal in-tuneness is required to bring about an energy difference sufficient to support perception. Beyond showing the shapes of the energetic valleys (of consonance) and peaks (of dissonance) he doesn't go further and quantify out-of-tuneness the way A. G. Bell investigated JND--"just noticeable differences"--in loudness that led to the deciBel.

I have experienced the difficulty of performing octaves in tune compared to perfect fifths. I both sing a cappella and play 'cello. I learned, once and suddenly, to my chagrin: don't accompany your own voice with octave doubling on sustained notes. You'll hear the problem too late.

Another practical example of Helmholtz' demonstration is well-known. We 'cellists and violists (and recently also arco bassists tuning in fifths) regularly and reluctantly tune our (lowest) C-string artificially sharp to accommodate a conventionally tuned keyboard instrument (octaves true). This entails tuning our (next-lowest) G-string, at second remove from A, also fudged narrow versus the D. Violinists, of course, have only one string to tune at more than one P5 remove from A.

To me, this goes far to make clear the practical value of the keyboard tuning procedure of "stretching the octaves" after the temperament has been set in a middle range of somewhat under two octaves.

An independent psychoacoustic observation, one that tends in the same direction (and the one cited as the rationale for stretching the octaves) is that mathematically exact octave multiples from a medial note are perceived out-of-tune "narrow", the more so as more distant octaves are sounded against it, whether high or low.

Given these same-tending and well-attested observations, why base a temperament on precise octaves? Why not on the P5 in its pythagorean 3:2? I've been tuning pianos for years on such a temperament, with solid client satisfaction. Let the fifths stretch the octaves.

Yes indeed, the P4 suffers. But we don't use it as scaffolding for chords. Even the suspended chord's P4 twirls, only temporarily, with her hand above her head in a major second brush to that of the P5 as he stands planted. She twirls (high energy) until, inevitably, the twirl resolves (lower energy) to thirds against root and fifth.

Very interesting posting, though I can't tell how I feel about some of it. However, I must say that this is brilliantly true, in my experience:

I assume that the human tendency to prefer harmonic intervals is a seamless instance of a widely general tendency, in us and other animals, to find lowest-energy solutions to hundreds of bodily tasks.

To give a musical example, I once had a church choir member, amateur, who developed an idiosyncratic bend of extreme vowels to a cheated value, somewhere between shwa and "ah". This was no expertly cheated high note by an accomplished soprano. I guessed this contralto sensed how she could sustain her invented diphthong with less energy--more comfortably, with more inner satisfaction. The result, repeated at the conclusion of every litany, every Sunday, was "To-ah thee-ah, O-ah Lo-ahd", with smooth but glaring glides.